aa r X i v : . [ m a t h . G T ] J a n EQUIVARIANT TORUS SIGNATURE AND PERIODIC RHO INVARIANT
LANGTE MA
Abstract.
Given an essentially embedded torus in a homology S × S , Echeverria [Ech19]and Ruberman [Rub20] defined two equivariant signature invariants from gauge theory andgeometric topology respectively as generalizations of the Levine-Tristram signature for knots.We prove the equivalence of these two invariants by introducing the periodic rho invariantassociated to the ASD DeRham complex over a cohomology T × S as a bridge. We alsoprove a surgery formula for the torus signature. Introduction
The equivariant signature invariant for knots and links were introduced by Levine [Lev69]and Tristram [Tri69] as concordance invariants. This invariant was also known to be relatedto the eta invariant of the twisted signature operator and twisted Casson invariant [Her97].For a nice summary of the properties and applications of the Levine-Tristram invariant, onemay consult the survey written by Conway [Con19]. Inspired by different perspectives towardsthe Levine-Tristram invariant, lately two authors considered four dimensional analogues withrespect to essentially embedded tori in homology S × S . The first approach is due to Echeverria[Ech19] by counting degree zero singular instantons. The second is due to Ruberman [Rub20]using a more topological construction. Conjecturally the signature invariants defined via bothmethods should agree. In this article, we give an affirmative answer to this conjecture. Thestrategy is to introduce a third type of invariants, which we call periodic rho invariants, as thebridge connecting these two invariants.To state the main results, we first review the mathematical objects and notations encounteredin this article briefly. The rigorous definitions will be distributed to the corresponding sections.Throughout this article, we will write V for a smooth closed oriented -manifold satisfying(1.1) H ∗ ( V ; Z ) ≃ H ∗ ( T × S ; Z ) . We refer to such a -manifold as a cohomology T × S . We fix a primitive class V ∈ H ( V ; Z ) represented by a smooth function f : V → S . The anti-self-dual DeRham operator is given by(1.2) Q := − d ∗ ⊕ d + : L ( T ∗ V ⊗ C ) −→ L ( V, C ) ⊕ L (Λ + T ∗ V ⊗ C ) . Given an admissible unitary representation ϕ : π ( V ) → U (1) in the sense of Definition 2.4,one gets the twisted ASD DeRham operator Q ϕ related to the covariant derivatives of the flatconnection corresponding to ϕ . Following the idea in the periodic version of Atiyah-Patodi-Singer theory [MRS16], one can define the periodic rho invariant of Q ϕ , denoted by ˜ ρ ϕ ( V, Q ) .We will show that ˜ ρ ϕ ( V, Q ) is a topological invariant of V only depending on the choice of theprimitive class V and the unitary representation ϕ . We shall write X for a smooth closed oriented -manifold satisfying(1.3) H ∗ ( X ; Z ) ≃ H ∗ ( S × S ; Z ) , which we refer to as an integral homology S × S . We also fix a primitive class X ∈ H ( X ; Z ) serving the role of homology orientations in Yang-Mills theory. Let T be an embedded torus in X inducing a surjective map H ( T ; Z ) → H ( X ; Z ) . We call T an essentially embedded torus.Given a holonomy parameter α ∈ (0 , / , adopting the frame work of α -singular instantonsin [KM93] Echeverria [Ech19] defined the α -singular Furuta-Ohta invariant λ F O ( X, T , α ) to bethe signed count of the degree zero irreducible α -singular instantons on ( X, T ) when α satisfiesa non-degeneracy condition Definition 4.2. Roughly speaking, an α -singular instanton is ananti-self-dual connection on X \T whose holonomy around the meridians of T is asymptotic to e − πiα ∈ U (1) ⊂ SU (2) . The non-degeneracy condition for α is equivalent to the fact that the reducible locus of the α -singular moduli space is isolated from the irreducible ones. With another non-degeneracycondition (4.16), one can define the usual Furuta-Ohta invariant λ F O ( X ) as a quarter of thecount of degree zero instantons. Echeverria proposed that the difference(1.4) λ F O ( X, T , α ) − λ F O ( X ) is an invariant for the isotopy class of essential embeddings of a -torus, and showed this differ-ence is the Levine-Tristram signature invariant σ α ( Y, K ) in the product case when ( X, T ) = S × ( Y, K ) . Here K is a knot in an integral homology sphere Y .On the other hand, given a pair ( X, T ) as above, one can perform the -surgery along thetorus T resulting in a cohomology T × S , which we denote by V . Moreover the primitive class X induces a primitive class V ∈ H ( V ; Z ) . Let Y V be a connected -manifold in V representingthe Poincaré dual of V . The holonomy parameter α gives rise to a unitary representation ϕ α : π ( V ) → U (1) mapping the meridian of T to e − πiα and the curve dual to V to . Wewrite ϕ ′ α = ϕ α | π ( Y V ) . Ruberman showed that the rho invariant introduced by Atiyah-Singer[AS68], denoted by ρ ϕ ′ α ( Y V ) , is independent of the choice of Y V when α = q/p r with p a primenumber. Using the fact that ρ ϕ ′ α ( Y V ) is a locally constant function with finite many jumpsrelative to varying α ∈ (0 , , up to averaging at the jumps, Ruberman defined the signatureinvariant of ( X, T ) twisted by α to be(1.5) σ α ( X, T ) := ρ ϕ ′ α ( Y V ) . When ( X, T ) is the product S × ( Y, K ) , the equality σ α ( X, T ) = σ α ( Y, K ) follows from thewell-known property of the Levine-Tristram signature. We use α here purely to match up thenormalization of the holonomy parameter in singular instantons, as the parameter in the rhoinvariant varies in (0 , .Our first result relates the periodic rho invariant with the difference (1.4). Theorem 1.1.
Let
T ⊂ X be an essentially embedded torus in a homology S × S satis-fying (4.16). Assume the holonomy parameter α ∈ (0 , / is non-degenerate in the sense ofDefinition 4.2. Then λ F O ( X, T , α ) − λ F O ( X ) = ˜ ρ ϕ α ( X ( T ) , Q ) , QUIVARIANT TORUS SIGNATURE AND PERIODIC RHO INVARIANT 3 where X ( T ) is the -surgered manifold of X along T , ϕ α : π ( X ( T )) → U (1) the uni-tary representation defined as above, and ˜ ρ ϕ α the periodic rho invariant of the ASD DeRhamoperator. Technically the definition of λ F O ( X, T , α ) requires using an orbifold metric with cone angle π/ν along T for some ν sufficiently large. Echeverria [Ech19] conjectured that the invariant isindependent of the choice of the cone angle. The proof of Theorem 1.1 also confirms this claim.Let’s call an embedded connected -manifold Y X ⊂ X a cross-section if [ Y X ] = PD 1 X ∈ H ( X ; Z ) . The second result relates the periodic rho invariant with the torus signature definedin (1.5). Theorem 1.2.
With the assumption of Theorem 1.1, we have ˜ ρ ϕ α ( X ( T ) , Q ) = σ α ( X, T ) . We will prove Theorem 1.2 with a further assumption that the representation ϕ α : π ( V ) → U (1) is admissible in the sense of Definition 2.4. Now we explain why such an assumption doesnot weaken the statement. From the construction, we see that both the periodic rho invariant ˜ ρ ϕ α ( X ( T ) , Q ) and the torus signature σ α ( X, T ) are locally constant as we vary α ∈ (0 , / .As illustrated in [GL16], the jumps of the Levine-Tristram signature of a knot correspond to theroots of the Alexander polynomial. The admissibility condition Definition 2.4 is meant to avoidthe jumps of ˜ ρ ϕ α ( X ( T ) , Q ) and σ α ( X, T ) . Since there are only finitely many jumps, localconstancy implies that both invariants share the same jumps. We will see in Lemma 5.1 that thenon-degeneracy condition in Definition 4.2 for the holonomy parameter α is equivalent to thefirst condition of the admissibility in Definition 2.4. As we mentioned above, the well-definednessof the singular Furuta-Ohta invariant λ F O ( X, T , α ) requires the non-degeneracy assumption for α . The jumping points are avoided automatically. Due to this reason, we believe (a) and (b) inthe admissibility condition Definition 2.4 are equivalent to each other in the case of cohomology T × S , and one should be able to argue in an elementary way using twisted cohomology. Toidentify (a) and (b) in Definition 2.4 via comparing both invariants, we need be more careful todeduce that there are no fake jumps given by both conditions, namely degenerate parameters α that give rise to the same signature invariant σ α ( X, T ) as that of nearby parameters. Thisis out of our pursuit in this article.With Theorem 1.1 and Theorem 1.2 we establish the equivalence between the two kinds ofsignature invariants for embedded tori in homology S × S . Corollary 1.3.
With the assumption of Theorem 1.1, we have λ F O ( X, T , α ) − λ F O ( X ) = σ α ( X, T ) . The equivalence of the signature invariants in Corollary 1.3 can be justified in the case ofmapping tori. Let’s consider a knot
K ⊂ Y in a integral homology sphere. Fix a positive integer n , we denote by Σ n ( Y, K ) the n -fold cyclic cover of Y branched along K , and τ n : Σ n ( Y, K ) → Σ n ( Y, K ) the covering translation. [LRS20, Proposition 6.1] tells us that the mapping torus X n ( Y, K ) of Σ n ( Y, K ) under τ n is an integral homology S × S . When Σ n ( Y, K ) is a rationalhomology sphere, X n ( Y, K ) satisfies the non-degenerate condition (4.16) automatically. Denoteby K n ⊂ Σ n ( Y, K ) the preimage of K in the branched cover, and T n ⊂ X n ( Y, K ) the mappingtorus of K n under τ n . LANGTE MA
Proposition 1.4.
With the notations as above, let α ∈ (0 , / be a holonomy parameter sothat e − πi ( α + j ) /n is not a root of the Alexander polynomial ∆ ( Y, K ) for each j = 0 , ..., n − . Then σ α ( X n ( Y, K )) = − n − X j =1 σ j/n ( Y, K ) + n − X j =0 σ α + j ) /n ( Y, K ) , where σ k/l ( Y, K ) is the Levine-Tristram signature of the pair ( Y, K ) . We note that in [Rub20] Ruberman computed the signature invariant σ α ( X n ( Y, K )) when α = q/p r with p prime. It’s straightforward to see that our computation agrees with his result[Rub20, Theorem 4.1] after unraveling the notations. However our proof goes through a differentpath by computing the singular Furuta-Ohta invariant. It was computed in [LRS20] that(1.6) λ F O ( X n ( Y, K )) = nλ ( Y ) + 18 n − X j =1 σ j/n ( Y, K ) , where λ ( Y ) is the Casson invariant of Y . As pointed out in [Ech19, Section 8], the assumptionof α in Proposition 1.4 implies the non-degeneracy condition in Definition 4.2. It follows fromCorollary 1.3, all we need is to derive(1.7) λ F O ( X, T , α ) = 8 nλ ( Y ) + n − X j =0 σ α + j ) /n ( Y, K ) . As communicated between the author and Echeverria, a correction of [Ech19, Theorem 46] willappear in a forthcoming version to establish the equality (1.7). For the sake of completeness,we supply a sketch of proof in subsection 5.3 following the original proof of Echeverria’s article.We also deduce a surgery formula for the singular Furuta-Ohta invariant. Given a pair ( X, T ) as before, one can perform /q surgery along the torus for an integer q by gluing the tubularneighborhood ν ( T ) with another boundary diffeomorphism to its complement. We denote theresulting manifold by X /q ( T ) , and the core of ν ( T ) = D × T in X /q ( T ) by T q . We notethat X /q ( T ) is still a homology S × S with T q essentially embedded. In the case of -surgery,we still denote by T the core of the gluing copy of D × T . Although X ( T ) is a homology T × S , we can still count α -singular instantons of the pair ( X ( T ) , T ) , which gives us aninvariant denoted by λ F O ( X ( T ) , T , α ) . Theorem 1.5.
Let α ∈ (0 , / be a non-degenerate holonomy parameter for the three pairs ( X, T ) , ( X /q ( T ) , T q ) , and ( X ( T ) , T ) . Then we have λ F O ( X /q ( T ) , T q , α ) = λ F O ( X, T , α ) + qλ F O ( X ( T ) , T , α ) . Combining the surgery formula for the Furuta-Ohta invariant [Ma20, Theorem 1.2], we geta surgery formula formula for the torus signature.
Corollary 1.6.
Suupose α ∈ (0 , / satisfies the assumptions in Theorem 1.1 and Theorem 1.5.Then we have σ α ( X /q ( T ) , T q ) − σ α ( X, T ) = qλ F O ( X ( T ) , T , α ) − qD w T ( X ( T )) , QUIVARIANT TORUS SIGNATURE AND PERIODIC RHO INVARIANT 5 where D w T ( X ( T )) is the count of irreducible ASD SO (3) -instantons on the SO (3) -bundle over X ( T ) with p = 0 and w = PD[ T ] . Outline.
We outline the content of this article. In Section 2, we define the periodic rho invari-ant of a cohomology T × S and prove it’s a topological invariant. In Section 3, we show howto extend the notion of periodic spectral flow in [MRS11] to ASD DeRham operators and provecorresponding properties. In Section 4, we give an alternative interpretation of the singularFuruta-Ohta invariant using results in [Ma20] and prove the surgery formula. In Section 5, weprove the main results of this article. Acknowledgment.
The author would like to thank Daniel Ruberman, Nikolai Saveliev, andMariano Echeverria for helpful discussions.2.
Periodic Rho Invariant
Let V be a cohomology T × S with a fixed primitive class V ∈ H ( X ; Z ) represented bya smooth function f : V → S , i.e. [ f ∗ dθ ] = 1 V with θ the arc-length parameter on the unitcircle. Let ˜ V be the infinite cyclic cover of V corresponding to V , and ˜ f : ˜ V → R a lift of f .We write df for the real -form on V descended from d ˜ f .2.1. The Periodic Eta Invariant.
Consider the L completion of the anti-self-dual part of the complexified DeRham complex:(2.1) −→ L ( V, C ) − d −−→ L ( V, T ∗ V ⊗ C ) d + −−→ L ( V, Λ + T ∗ V ⊗ C ) −→ . Let’s choose a branch of the logarithm map to define z w = e w ln z for z ∈ C ∗ , w ∈ C . We getthe twisted anti-self-dual DeRham complex:( E z ) −→ L ( V, C ) − d z −−→ L ( V, T ∗ V ⊗ C ) d + z −−→ L ( V, Λ + T ∗ V ⊗ C ) −→ , where d z = d − ln z · df , d + z = d + − ln z · ( df ∧ − ) + . We note that the sequence ( E z ) is elliptic inthe sense that d + z ◦ d z = 0 and the corresponding sequence of symbols is exact. Thus we havewell-defined cohomology H ∗ ( E z ) . An adaptation of [Tau87, Theorem 3.1] implies the followingproperty of ( E z ). Lemma 2.1.
The cohomology of the complex ( E z ) vanishes for all z ∈ C ∗ but a discrete setwith no accumulation points.Proof. Due to [Tau87, Lemma 3.2], one only needs to check that − b + b + = 0 for V and df ∧ : H DR ( V ) → H DR ( V ) has -dimensional kernel. Both conditions are automatic due to ourassumption that V is a cohomology T × S . (cid:3) Definition 2.2.
We call z ∈ C ∗ a regular point of the anti-self-dual DeRham complex over V if H ∗ ( E z ) = 0 . Otherwise, z is called a spectral point. We denote by Σ( E z ) the set of spectralpoints of ( E z ). LANGTE MA
Rather than keeping track of the complex ( E z ), we can consider the equivalent elliptic oper-ator(2.2) Q z := − d ∗ z ⊕ d + z : L ( T ∗ V ⊗ C ) −→ L ( V, C ) ⊕ L (Λ + T ∗ V ⊗ C ) , where d ∗ z = d ∗ − ln ¯ z · ι df is the L -adjoint of d z and ι df is the contraction with df . Then wehave(2.3) dim ker Q z = dim H ( E z ) , dim coker Q z = dim H ( E z ) + dim H ( E z ) , which tells us that z is a regular point of ( E z ) if and only if Q z is invertible. We will simplywrite Q = Q = − d ∗ ⊕ d . The L -adjoin of Q z is given by(2.4) Q ∗ z = − d z ⊕ d ∗ z : L ( V, C ) ⊕ L (Λ + T ∗ V ⊗ C ) −→ L ( T ∗ V ⊗ C ) . The periodic eta invariants of the ASD DeRham complex associated to the pair ( V, V ) isdefined as follows similar to the non-Fredholm case for Dirac operators in [MRS16, Section 7].We note that ker Q = H ( V ; C ) = 0 . Thus z = 1 is a spectral point. Due to Lemma 2.1 onecan choose ǫ > small enough so that(2.5) Σ( E z ) ∩ { z : e − ǫ < | z | < e ǫ } ⊂ { z : | z | = 1 } , where Σ( E z ) is the set of spectral points of ( E z ). Given = δ ∈ ( − ǫ, ǫ ) , we define(2.6) η δ ( V ) = 1 πi Z ∞ I | z | = e δ Tr ( df · Q z exp( − tQ ∗ z Q z )) dzz dt, where df · is applied to L k − ( V, C ) ⊕ L k − (Λ + T ∗ V ⊗ C ) by df · ( s, η ) = ( sdf, ι df η ) . We willsee shortly in the proof of Theorem 2.3 that η δ ( V ) is independent of the choice of the sign of δ ∈ ( − ǫ, ǫ ) \{ } . Thus we will simply denote by η ± ( V ) the corresponding periodic eta invariantsgiven by positive and negative δ respectively. Finally we take the mean of the eta invariantsand denote it by(2.7) η ( V ) := η + ( V ) + η − ( V )2 . One can interpret the periodic eta invariants as the spectral asymmetry of the complex ( E z ).Explicitly, it’s the regularization of the difference between the number of spectral points in Σ( E z ) of length greater and less than . See [MRS16, Section 6] for more details of this inter-pretation in the case of Dirac type operators.2.2. Relation with End-Periodic Operators.
In [MRS16], Mrowka-Ruberman-Saveliev generalized the result of the index theorem ofAtiyah-Patodi-Singer [APS75a] to Dirac type operators over end-periodic manifolds. It turnsout their result also applies for the ASD DeRham operator Q . Let Y ⊂ V be an embeddedconnected -manifold representing the Poincaré dual of V ∈ H ( V ; Z ) . Let W be the cobor-dism from Y to itself obtained by cutting V along Y . The cyclic cover of V can be constructedas ˜ V = S n ∈ Z W n , where each W n is a copy of W . Let ˜ V + = S n ≥ W n be the non-negative halfof the cyclic cover. Picking a compact -manifold M with ∂M = Y , we form an end-periodic QUIVARIANT TORUS SIGNATURE AND PERIODIC RHO INVARIANT 7 manifold Z = M ∪ ˜ V + . Let τ : Z → R be a smooth extension of ˜ f | ˜ V + . The weighted Sobolevspace with weight δ of function spaces over Z is defined to be L k,δ := e − δτ L k .We can equally consider the ASD DeRham complex over Z with weight δ > :( E δ ( Z ) ) −→ L ,δ ( Z, C ) − d −−→ L ,δ ( Z, T ∗ Z ⊗ C ) d + −−→ L δ ( Z, Λ + T ∗ Z ⊗ C ) −→ . The criterion given by Taubes [Tau87, Lemma 4.3] asserts that ( E δ ( Z ) ) is Fredholm if andonly if ( E z ) has vanishing cohomology for all z with | z | = e δ . Since z = 1 is a spectral point of( E z ), we need to work with weights δ = 0 to get a Fredholm complex ( E δ ( Z ) ) with well-definedindex. Conjugating by the weight e δτ , ( E δ ( Z ) ) is equivalent to(2.8) −→ L ( Z, C ) − d δ −−→ L ( Z, T ∗ Z ⊗ C ) d + δ −−→ L ( Z, Λ + T ∗ Z ⊗ C ) −→ , where d δ = e δτ de − δτ = d − δdτ , and d + δ = d + − δ ( dτ ∧ − ) + . Let d ∗ δ = d ∗ − δ · ι dτ be the L -adjoint of d δ . We let Q δ ( Z ) = − d ∗ δ ⊕ d + δ be the operator over Z on L k completedspaces. The L -adjoint of Q δ ( Z ) is given by Q ∗ δ ( Z ) = ( − d + δ · dτ, d ∗ − δ · ι dτ ) . When the context is clear, we will simply write Q δ and Q ∗ δ To relate the index of Q δ with η δ ( V ) , we follow the strategy of [MRS16] by considering thetwisted operators Q δ,z and Q ∗ δ,z on V associated to z ∈ C ∗ . Let a ∈ C ∞ ( ˜ V , T ∗ ˜ V ⊗ C ) be acompactly supported complex -form over the infinite cyclic cover of V . The Fourier-Laplacetransform [MRS16] of a with respect to z ∈ C ∗ is defined as(2.9) ˆ a z ( x ) = z ˜ f ( x ) ∞ X n = −∞ z n a ( x + n ) . Let’s write x +1 for the point in ˜ V obtained by applying the generating covering transformationto x once. Then ˆ a z ( x + 1) = ˆ a z ( x ) . Thus we get a complex -form ˆ a z on V . Since the Fourier-Laplace transform is invertible, we define the twisted operator Q δ,z on V via the formula: Q δ,z (ˆ a z ) = \ Q δ ( a ) z . Similarly Q ∗ δ,z is obtained from Q ∗ δ via conjugating the Fourier-Laplace transform. More explic-itly, we have Q δ,z = (cid:0) − d ∗ + ( δ − ln z ) ι df , d + − (ln z + δ )( df ∧ − ) + (cid:1) Q ∗ δ,z = ( − d + ( δ + ln z ) df, d ∗ − ( δ − ln z ) ι df ) (2.10)We choose = δ ∈ ( − ǫ, ǫ ) with ǫ satisfying (2.5). Then the operator Q δ is Fredholm byTaubes’ criterion [Tau87, Lemma 4.3]. Since Q is an elliptic first order differential operator, itfollows from the Atiyah-Singer index theory that Q admits an index form denoted by a ( Q ) .Since ind Q ( V ) = 0 , this index form is exact on V for which we write as dω = a ( Q ) . Foreach z ∈ Σ( E z ) , one can assign a positive integer d ( z ) as the multiplicity of z as in [MRS16,(33)]. Then the main result [MRS16, Theorem C] in their article applies to our case. Note that LANGTE MA ind Q δ ( Z ) is independent of the choice of δ ∈ (0 , ǫ ) . Thus we denote it by ind + Q ( Z ) . Similarly,we write ind − Q ( Z ) for ind Q δ ( Z ) for δ ∈ ( − ǫ, . Theorem 2.3 ([MRS16, Theorem C]) . With the notations above, we have ind + Q ( Z ) = Z M a ( Q ) − Z Y ω + Z V df ∧ ω − h + η ( V )2 , where h = P z d ( z ) with the sum over Σ( E z ) ∩ { z : | z | = 1 } .Proof. The proof is literally the same as that of [MRS16, Theorem C] except for some minoradjustment. In our case the corresponding symmetric trace takes the form
Str ♭ ( δ, t ) = Tr ♭ exp( − tQ ∗ δ Q δ ) − Tr ♭ exp( − tQ δ Q ∗ δ ) , where Tr ♭ means the regularized trace of an elliptic operator over the end-periodic manifold Z defined in [MRS16, Section 3]. Since the differential operator Q ∗ Q = d ∗ d + d ∗ d + is positiveself-adjoint and elliptic, the estimates of smoothing kernels in [MRS16] can be carried over to Q ∗ Q as well, which leads to the following limits: lim t →∞ Str ♭ ( δ, t ) = ind Q δ ( Z ) , lim t → Str ♭ ( δ, t ) = Z M a ( Q ) . In the meantime, the formula [MRS16, (23)] corresponding to Q is ddt Str ♭ ( δ, t ) = 12 πi I | z | =1 Z W f · tr (cid:16) K exp( − tQ ∗ δ,z Q δ,z ) − K exp( − tQ δ,z Q ∗ δ,z ) (cid:17) dx dzz − πi I | z | =1 Tr (cid:18) ∂Q ∗ δ,z ∂z · Q δ,z exp( − tQ ∗ δ,z Q δ,z ) (cid:19) dz. (2.11)We note that ∂Q ∗ δ,z /∂z = 1 /z ( df, ι df ) and when | z | = 1 , we have Q δ,z = Q e δ z and Q ∗ δ,z = Q ∗ e δ z . Thus η δ ( V ) / can be identified with the integration of the second term of (2.11). Then the restof the proof follows from the discussion in [MRS16, Section 7.3]. (cid:3) The Periodic Rho Invariant.
Let ϕ : π ( V ) → U (1) be a U (1) -representation of the fundamental group. The covariantderivative of the flat connection on the trivial bundle corresponding to ϕ is denoted by d ϕ .Then for z ∈ C ∗ , we have the ASD twisted DeRham complex:( E ϕ,z ) −→ L ( V, C ) − d ϕ,z −−−→ L ( V, T ∗ V ⊗ C ) d + ϕ,z −−→ L ( V, Λ + T ∗ V ⊗ C ) −→ defined as before. Let Q ϕ = − d ∗ ϕ ⊕ d + ϕ . Definition 2.4.
The representation ϕ : π ( V ) → U (1) is admissible if the following holds:(a) For all z with | z | = 1 , H ∗ ( E ϕ,z ) = 0 .(b) There exists a cross-section Y ⊂ V such that H ( Y, C ϕ ) = H ( Y, C ϕ ) = 0 . QUIVARIANT TORUS SIGNATURE AND PERIODIC RHO INVARIANT 9
The admissibility condition in Definition 2.4 corresponds to that in Taubes’ article [Tau87,Definition 1.3]. The first condition (a) will be seen to coincide with the non-degeneracy conditionDefinition 4.2 for the moduli space of singular instantons. Meanwhile, (a) also implies that thespectral set Σ( E ϕ,z ) of the complex ( E ϕ,z ) is discrete via the argument of [Tau87, Lemma 4.5].The second condition (b) corresponds to the fact that the rho invariant ρ ϕ ( Y ) is not a jumpingpoint with respect to varying the representation ϕ .With the admissible assumption, we can proceed as above to define h ( Q ϕ ) as the count of Σ( E ϕ,z ) ∩ { z : | z | = 1 } with multiplicity and η ( Q ϕ ) as the spectral asymmetry of Q ϕ . We let ρ ( V, Q ϕ ) = h ( Q ϕ ) + η ( Q ϕ )2 . The periodic rho invariant is defined to be(2.12) ˜ ρ ϕ ( V, Q ) = ρ ( V, Q ϕ ) − ρ ( V, Q ) . Lemma 2.5.
The periodic rho invariant ˜ ρ ϕ ( V, Q ) is independent of the choice of the smoothfunction f : V → S and the Riemannian metric g on V .Proof. The independence of f follows from the argument of [MRS16, Lemma 8.2] directly. Tosee the independence on the metric, we adopt the argument in [MRS16, Lemma 8.3]. Let g and g be two metrics on V . We choose a metric ˜ g and a function ˜ f ′ on ˜ V = S n ∈ Z W n so that ˜ g | W n = ( g if n < g if n > , ˜ f ′ | W n = ( − ˜ f if n < f if n > . We regard ( ˜
V , ˜ g, ˜ f ′ ) as an end-periodic manifold with two periodic ends attached to ( W , ˜ g | W ) .Then Theorem 2.3 implies that ind + Q ϕ ( ˜ V , ˜ g ) − ind + Q ( ˜ V , ˜ g ) = ˜ ρ ϕ ( Q, g ) − ˜ ρ ϕ ( Q, g ) + h ( Q, g ) − h ( Q ϕ , g ) . By letting g = g and ˜ g = ˜ g the product metric, we know h ( Q, g ) − h ( Q ϕ , g ) = ind + Q ϕ ( ˜ V , ˜ g ) − ind + Q ( ˜ V , ˜ g ) . Thus we get ˜ ρ ϕ ( g ) − ˜ ρ ϕ ( g ) = (cid:16) ind + Q ϕ ( ˜ V , ˜ g ) − ind + Q ( ˜ V , ˜ g ) (cid:17) − (cid:16) ind + Q ϕ ( ˜ V , ˜ g ) − ind + Q ( ˜ V , ˜ g ) (cid:17) . So it suffices to show that ind + Q ϕ ( ˜ V , ˜ g ) is independent of the metric with respect to anyadmissible ϕ : π ( V ) → U (1) .Let g t , t ∈ [0 , , be a path of metrics connecting g and g . This path can be lifted, afterappropriate modification on W , to a path of metrics ˜ g t on ˜ V from ˜ g to ˜ g . Once we prove thatone can find δ > such that Q ϕ ( ˜ V , ˜ g t ) is a Fredholm operator on L k,δ for all t ∈ [0 , . Theconclusion then follows from the invariance of index under continuous deformation of Fredholmoperators.Taubes’ criterion tells us that Q ϕ,δ ( ˜ V . ˜ g t ) is Fredholm if and only if H ( E ϕ,z ( g t )) = 0 for all | z | = e δ . Since d ϕ,z = 0 , d ϕ,z can be regarded as the covariant derivative given by a flat con-nection (not necessarily unitary) on the trivial bundle C of V . The Chern-Weil homomorphismimplies that d ϕ,z a = 0 once d + ϕ,z a = 0 . Thus H ( E ϕ,z ( g t )) ≃ H ( V, C ϕ z ) with ϕ z : π ( V ) → C ∗ the representation given by the holonomy of the flat connection d ϕ,z , which is clearly indepen-dent of the metric. (cid:3) Index Computation.
Let Z = M ∪ ˜ V + be an end-periodic manifold as in subsection 2.2 and ϕ : π ( V ) → U (1) arepresentation. Since π ( ˜ V ) → π ( V ) is injective, we get a unitary representation ˜ ϕ : π ( ˜ V ) → U (1) by restriction. Suppose we can extend this representation to one over Z , which we stilldenote by ˜ ϕ : π ( Z ) → U (1) . We further assume that ˜ ϕ is trivial on the torsion part of π ( Z ) .Then ˜ ϕ corresponds to a flat connection on the trivial bundle C whose covariant derivative isdenoted by d ˜ ϕ . We can consider the complex( E ˜ ϕ,δ ( Z ) ) −→ L ,δ ( Z, C ) − d ˜ ϕ −−→ L ,δ ( Z, T ∗ Z ⊗ C ) d +˜ ϕ −−→ L δ ( Z, Λ + T ∗ Z ⊗ C ) −→ . The following identification of the cohomology is an adaptation of the corresponding result inTaubes’ article.
Proposition 2.6 ([Tau87, Proposition 5.1]) . Let ϕ : π ( V ) → U (1) be an admissible represen-tation. Then there exists δ > such that for all δ ∈ (0 , δ ) the complex ( E ˜ ϕ,δ ( Z ) ) is Fredholmwith cohomology given by H ( E ˜ ϕ,δ ( Z )) = 0 , H ( E ˜ ϕ,δ ( Z )) = H ( M, C ˜ ϕ ) , H ( E ˜ ϕ,δ ( Z )) = H + ( M, C ˜ ϕ ) . Proof.
Let’s write d ˜ ϕ = d + a with a an end-periodic harmonic i R -valued -form over Z . Since H ( E ˜ ϕ,δ ( Z )) consists of ˜ ϕ -invariance functions of exponential decay, only the constant zerofunction lies here.To identify H ( E ˜ ϕ,δ ( Z )) , ω ∈ ker d +˜ ϕ | L ,δ . Since ω has exponential decay, Stokes’ formulagives us Z Z d ( ω ∧ d ˜ ϕ ω ) = Z Z | d +˜ ϕ ω | − Z Z | d − ˜ ϕ ω | . So we conclude that d ˜ ϕ ω = 0 . Combining (a) and (b) in Definition 2.4, we see that H ( ˜ V + , C ϕ ) =0 . Thus ω | ˜ V + = d ˜ ϕ s + for some s + ∈ C ∞ ( ˜ V + , C ) . The rest follows from the argument in [Tau87,p.382].Note that H ( E ϕ ) = H + ( V, C ϕ ) = 0 . The last assumption in [Tau87, Definition 1.3] issatisfied. The argument in [Tau87, Lemma 5.7] then carries through to give us the desiredidentification. (cid:3) Periodic Spectral Flow
The periodic spectral flow was first introduced by Mrowka-Ruberman-Saveliev [MRS11] forDirac operators over homology S × S , and later generalized by the author [Ma19] to Diracoperators over manifolds with cylindrical ends. The goal of this section is to recover such anotion for a path of ASD DeRham operators Q t as we have considered above. We first considerthe periodic spectral flow over a cohomology T × S , then derive a corresponding notion overmanifolds having the same homology of T × D and a cylindrical end modeled on [0 , ∞ ) × T . QUIVARIANT TORUS SIGNATURE AND PERIODIC RHO INVARIANT 11
The Definition.
Let V be a cohomology T × S with a choice of smooth function f : V → S as before. Wedenote by A k ( V, C ) the space of L k unitary connections on the trivial line bundle over V forsome fixed k ≥ . To better cooperate with Yang-Mills theory, we consider perturbations of theASD DeRham operator. Let P = L k (cid:0) X, ( T ∗ V ⊗ C ) ∗ ⊗ Λ + T ∗ V ⊗ C (cid:1) be the space of perturbations. Since k ≥ , we know π : L k ( T ∗ V ⊗ C ) → L k (Λ + T ∗ V ⊗ C ) is abounded linear map with norm given by k π k L k . Typical examples of such perturbations ariseas the differential of holonomy perturbations [Don87]. We are primarily interested in the casewhen the unitary connection A is flat. Given π ∈ P , the perturbed ASD DeRham operator isdenoted by(3.1) Q A ( V, π ) = − d ∗ A ⊕ d + A,π , where d + A,π = d + A + π . We note that as an operator from L k to L k − spaces, the perturbation π is compact. So ind Q A ( V, π ) = ind Q A ( V ) = 0 . Given z ∈ C ∗ , we can equally consider theoperator Q A,z ( V, π ) = − d ∗ A,z ⊕ d + A,z,π . We denote the spectral set by Σ( A, π ) := { z ∈ C ∗ : Q A,z ( V, π ) is not invertible } . When π = 0 , Σ( A,
0) = Σ( E z ) . Given δ ∈ R , let’s write S δ = { z ∈ C : | z | = e δ } for the circleof radius e δ . Definition 3.1.
We call a continuous path ( A t ) , t ∈ [0 , , of connections in A k δ -regular withrespect to a perturbation π ∈ P if the following holds.(i) When i = 0 , , the operator Q A i ,z ( V, π ) has trivial kernel with respect to all z ∈ C ∗ oflength e δ .(ii) The spectral set Σ δ ( A t , π ) := (cid:8) ( t, z ) ∈ [0 , × S δ : ker Q A t ,z ( V, π ) = 0 (cid:9) is discrete and ker Q A t ,z ( V, π ) = C , ∀ ( t, z ) ∈ Σ δ ( A t , π t ) . Lemma 3.2.
Given δ ∈ R , let ( A t ) be a path of flat connections in A k such that A t − ln z · df isnot the product connection for all t ∈ [0 , , z ∈ S δ . Then ( A t ) is a δ -regular path with respectto a generic perturbation π ∈ P .Proof. Let C [0 , δ = [0 , × S δ be the parameter space. Take a connection A = A t ,z for some ( t , z ) ∈ C [0 , δ . Denote by H = ker Q A and H = (im Q A ) ⊥ . Since Q A is Fredholm of index , we know dim H = dim H is finite. We consider the map η : P × C [0 , δ × ker d ∗ A −→ im d + A ( π, t, z, a ) Π d + A t,z ,π a, (3.2) where Π : L k − (Λ + T ∗ V ⊗ C ) → im d + A is the L -projection to the image of d + A . Since A isa flat connection, Hodge theory tells us that the differential Dη | (0 ,t ,z ,a ) is surjective on the ker d ∗ A component. Invoking the implicit function theorem, we can find a neighborhood U × U of (0 , t , z ) in P × C [0 , δ and a map h : U × U × H → ker d ∗ A such that for all ( π, t, z, a ) ∈ U × U × H we have η ( π, t, z, a + h ( π, t, z, a )) = 0 . Equivalently, d + A t,z ,π ( a + h ( π, t, z, a )) ∈ H .Since η is linear in a , so is h . In this way we get a map ξ : U × U −→ Hom C ( H , H )( π, t, z ) (cid:16) a d + A t,z ,π ( a + h ( π, t, z, a )) (cid:17) . (3.3)The differential of ξ at (0 , is Dξ | (0 , ( π,
0) : a π ( a ) + d + A ◦ Dh | (0 , ,a ) π . By the constructionwhen π ( H ) ⊂ H , we have Dξ | (0 , ( π,
0) = π ( a ) . Restricting to such perturbations, we concludethat ρ is a submersion at (0 , . Note that the stratum of linear maps in Hom C ( H , H ) withdimension- i kernels is of complex codimension i . Since dim R U = 2 , the Sard-Smale theoremtells us that for a generic perturbations dim C Π ′ ker d + A t,z ,π ≤ with equality only at a discreteset of points, where Π ′ is the projection to ker d ∗ A ∩ ker d + A = ker Q A .Note that the dimension of kernel is an upper semi-continuous function on Fredholm op-erators and the projection Π ′ : ker Q A → ker Q A is injective for A near A . By shrinking U possibly, we have proved that for each ( t , z ) one can find a neighborhood U of ( t , z ) in C [0 , δ over which the δ -regular condition is satisfied with respect to generic perturbations.Since C [0 , δ is compact, we can run the argument in finite steps to conclude. We also remarkthat since dim R ∂C [0 , δ = 1 , one can first run the argument there to arrive at the case that Q A is invertible for all A on the boundary. (cid:3) Given a δ -regular path A t with respect to π , we write ( t j , z j ) ∈ Σ δ ( A t , π ) for the spectralpoints on the cylinder. The following result in [MRS11] describes the spectral set in a neighbor-hood of the cylinder. Although the original statement is proved for Dirac operators, the proofonly uses the property of index zero Fredholm operators. So it also works in our case. Proposition 3.3. [MRS11, Theorem 4.8]
Let ( A t ) be a δ -regular path with respect to a pertur-bation π ∈ P . Then there exists ǫ > and ǫ > [ | t − t j | <ǫ { t } × (Σ( A t , β ) ∩ B ǫ ( z j )) ⊂ [0 , × C ∗ is a smoothly embedded curve for all j , where B ǫ ( z j ) is the open disk of radius ǫ centered at z j . We recall the definition of the periodic spectral flow introduced in [MRS11]. For more detailsand explanation of the following definition, one may consult [Ma19, Section 4]
Definition 3.4.
We say λ > is an excluded value for Q A ( V, π ) if Q A,z ( V, π ) is invertible for all z of length λ . Definition 3.5.
Let ( A t ) be a δ -regular path with respect to π . A system of excluded valuesfor Q A t ( X, π ) is a finite sequence of pairs { ( t l , λ l ) } nl =1 satisfying QUIVARIANT TORUS SIGNATURE AND PERIODIC RHO INVARIANT 13 (a) t < t < ... < t n = 1 is a partition of [0 , ;(b) λ l ∈ ( e δ − ǫ , e δ + ǫ ) is an excluded value for Q A t ( V, π ) for all t ∈ [ t l − , t l ] , where ǫ isthe constant in Proposition 3.3. Moreover λ = λ n = e δ . Definition 3.6.
Let { ( t l , λ k ) } nl =1 be a system of excluded values for a δ -regular path ( A t ) withrespect to π . For l = 1 , ..., n − , we define a l = if λ l > λ l +1 if λ l = λ l +1 − if λ l < λ l +1 and b l = { z ∈ Σ( A t l , π ) : min( λ l , λ l +1 ) ≤ | z | ≤ max( λ l , λ l +1 ) } . The periodic spectral flow of the family Q A t ( V, π ) along the δ -regular path ( A t ) is defined tobe e Sf( Q A t ( V, π )) := n − X l =1 a l b l ∈ Z . Remark 3.7.
It follows from [Ma19, Lemma 4.9] that the periodic spectral flow is independentof the choice of systems of excluded values.Given a path ( A t ) of flat connections satisfying the assumption of Lemma 3.2, [Ma19, Lemma4.10] implies that as long as the generic perturbations are small enough, the periodic spectralflow is independent of the choice of perturbations as well. In this case, we will simple write e Sf( Q A t ( V )) for the periodic spectral flow with respect to small perturbations.Just like the case for Dirac operators in [Ma19], we also consider periodic spectral flow definedover manifolds with boundary. The idea is to attach a cylindrical end and impose certain decayconditions for the connections, which enables us to treat the ASD DeRham operator associatedto these connections as if defined over closed manifolds.Let M be a smooth compact -manifold with ∂M = Y . For simplicity we assume Y is con-nected. We write M o = M ∪ [0 , ∞ ) × Y for the manifold obtained by attaching a cylindrical endto M . We assume that b ( M ) > . Then we fix a primitive class M ∈ H ( M o , Z ) representedby a function f : M o → S whose restriction on the end, f | { t }× Y = f Y for t ∈ [0 , ∞ ) , istime-independent.Any connection A on the trivial line bundle over M o takes the form A | [0 , ∞ ) × Y = dt ⊗ c ( t ) + B ( t ) with B ( t ) a time-dependent connection on Y and c ( t ) a time-dependent complex-valuedfunction on Y . Let I ⊂ [0 , ∞ ) be a sub-interval. Recall there is a canonical way to identifyforms on I × Y with time-dependent forms on Y :(1) Ω ( I × Y ) = C ∞ ( I, Ω ( Y )) via s s | { t }× Y ,(2) Ω ( I × Y ) = C ∞ ( I, Ω ( Y ) ⊕ Ω ( Y )) via a = dt ⊗ s ( t ) + b ( t ) (2 s ( t ) , b ( t )) .(3) Ω + ( I × Y ) = C ∞ ( I, Ω ( Y )) via ω = dt ∧ b ( t ) + ⋆b ( t ) b ( t ) . With these identifications, the ASD DeRham operator over the end [0 , ∞ ) × Y takes the form Q A = d/dt + c ( t ) + L B ( t ) with(3.4) L B = (cid:18) − d ∗ B − d B ⋆d B (cid:19) . We can further identify Ω ( Y ) with Ω ( Y ) by b
7→ − ⋆ b . Then L B is equivalent to the operatoracting on even forms: L ′ B : Ω p ( Y ; C ) −→ Ω p ( Y ; C ) ω ( − p +1 ( ⋆d B − d B ⋆ ) ω (3.5)Similarly we get Q A,z = d/dt + c ( t ) + L B ( t ) ,z with L B ( t ) ,z = L B − ln z · df Y . Definition 3.8.
We say a connection A on M o is δ -regularly asymptotically flat ( δ -RAF) isthe following hold:(a) There exists T > , µ > , and a flat connection B o on Y such that k B ( t ) − B o k C ( Y ) + k c ( t ) k C ( Y ) ≤ c e − µ ( t − T ) for some constant c > ;(b) The twisted operator L B o ,z : L → L is invertible for all z of length e δ ;(c) There exists ǫ Y > such that Q A,z ( M o ) : L → L has index for all z with length inthe range ( e δ − ǫ Y , e δ + ǫ Y ) .The definition requires some explanation. Assuming (a) in Definition 3.8, we have Q A,z | [ T, ∞ ) × Y = ddt + L B o ,z + L B ( t ) ,z − L B o ,z + c ( t ) . The assumption of exponential decay implies that the zero order operator over [ T, ∞ ) × Y , L B ( t ) ,z − L B o ,z + c ( t ) : L → L , is compact. Then the second assumption (b) implies that Q A,z ( M o ) is Fredholm for | z | = e δ . The openness of being Fredholm gives us the existence ofsuch ǫ Y to guarantee that Q A,z ( M o ) is Fredholm with respect to z in the asserted range. Therequirement of having index zero can be justified by the topology of M using the theory ofAtiyah-Patodi-Singer [APS75a].Over the non-compact manifold M o , we consider perturbations of exponential decay alongthe cylindrical end. For simplicity we may fix a weight µ > , and let(3.6) P µ = L k,µ (cid:0) M o , ( T ∗ M o ⊗ C ) ∗ ⊗ Λ + T ∗ M o ⊗ C (cid:1) be the space of perturbations. Then for each π ∈ P µ , we get a compact operator π : L k → L k − .The notion of δ -regularity in Definition 3.1 can be defined for a path ( A t ) of δ -RAF connectionsover M o due to the assumption (c) in Definition 3.8. The argument of Lemma 3.2 can be appliedverbatim to show that any path ( A t ) of flat δ -RAF connections is δ -regular with respect to ageneric perturbation π ∈ P µ . This leads us to define the periodic spectral flow of such a path. QUIVARIANT TORUS SIGNATURE AND PERIODIC RHO INVARIANT 15
Properties.
We shall demonstrate two properties of periodic spectral flow. The first is a gluing propertyrelating the notion over closed manifolds to manifolds with cylindrical end. The second is toestablish the relation with the periodic ˜ ρ invariant. Both properties are well-known for the usualspectral flow, and proved in the case of Dirac operators in [Ma19] and [MRS11] respectively.We restrict ourselves to a concrete model. Let X be a cohomology T × S decomposed alongan embedded -torus Y into two compact pieces: V = M ∪ Y N . We further assume that M isa cohomology D × T and N = D × T . Let g be a metric on V so that g | ( − , × Y = dt + h for a flat metric h on the -torus Y . Given T > , we write V T for the Riemannian manifoldobtained from V by inserting a "neck" of length T about Y . More specifically, we write M T = M ∪ [0 , T ] × Y , N T = [ − T, × Y ∪ N , and V = M T ∪ N T . The "neck" is identifiedwith [ − T, T ] × Y ⊂ V T . The limit of such metrics breaks V into two pieces, which we denoteby M o = M ∪ [0 , ∞ ) × Y and N o = ( −∞ , × Y ∪ N .We also fix a primitive class V ∈ H ( V ; Z ) as before, which restricts to primitive classeson M and N . We choose a representative function f : V → S satisfying f | ( − , × Y = f Y forsome function f Y : Y → S . Then the restriction of f can be extended to f M : M o → S and f N : N o → S taking the form of f Y on the cylindrical ends.Suppose for each T ∈ [0 , ∞ ) we have a path ( A T,s ) of flat connections over the trivial linebundle of X converging to δ -RAF paths ( A M,s ) and ( A N,s ) on M o and N o in the followingsense:(1) For each s ∈ [0 , , A M,s and A N,s are both asymptotic to the same flat connection B s over Y ;(2) For each s ∈ [0 , , A T,s | M T and A T,s | N T converges to A M,s and A N,s respectively as T → ∞ in the L k,loc topology;(3) Given any ǫ > , one can find T ǫ such that for all T > T ǫ we have k A T,s | [ T ǫ − T,T − T ǫ ] × Y − A B s k L k < ǫ, where A B s = dt + B s is the connection on [ T ǫ − T, T − T ǫ ] × Y .Moreover assume we can find perturbations π T ∈ P ( V T ) converging to π M ∈ P ( M o ) and ∈ P ( N o ) as T → ∞ so that the paths ( A T,s ) , ( A M,s ) , and ( A N,s ) are δ -regular with respectto the perturbations π T , π M , and respectively.The following result is drawn from [Ma19]. Although originally the author was consideringDirac operators, the proof goes through without any change. The idea is that when we imposethe condition that the path ( A N,s ) admits empty spectral curve near the δ -circle S δ , the spectralcurves of paths ( A T,s ) and ( A M,s ) can be identified when T ≫ . In general if ( A N,s ) contributesspectral curves, the author expects an additive formula of the periodic spectral curve as in thework of [CLM96] for usual spectral flows. Proposition 3.9 ([Ma19, Theorem 4.16]) . Under the situation as above. We further assumethat Q A N,s ,z ( N o ) is invertible for all ( s, z ) ∈ [0 , × S δ . Then there exists T o > such that forall T > we have e Sf( Q A T ,s ( V T , π T )) = e Sf( Q A M,s ( M o , π M )) . Now we continue to assume V to be a cohomology T × S with a function f : V → S representing a primitive class in H ( V ; Z ) . Let ϕ : π ( V ) → U (1) be an admissible unitaryrepresentation satisfying. Then we have a well-defined invariant ˜ ρ ϕ ( V, Q ) = ρ ( V, Q ϕ ) − ρ ( V, Q ) .Denote by A ϕ the flat connection on the trivial line bundle of V whose holonomy is ϕ . Let A be the product connection, and A t = A + t ( A ϕ − A ) be the path of flat connections from A to A ϕ . Then we know the family of connections A t,z = A + t ( A ϕ − A ) − ln zdf, | z | = e δ does not contain the product connection A whenever δ = 0 . Proposition 3.10.
Let ϕ and ( A t ) be chosen as above, and ǫ > be a positive constantsatisfying (2.5). Then for each δ ∈ (0 , ǫ ) , one can choose generic small perturbations π ∈ P turning ( A t ) into a δ -regular path so that − ˜ ρ ϕ ( V, Q ) = e Sf( Q A t ( V, π )) . Proof.
For fixed δ ∈ (0 , ǫ ) , we can choose generic perturbations π ∈ P to make ( A t ) a δ -regularpath appealing to Lemma 3.2.We recall the set-up in subsection 2.2 where we form a end-periodic -manifold Z = M ∪ ˜ V + with the period modeled on the cobordism W obtained by cutting V along a representative ofthe Poincaré dual of V ∈ H ( V, Z ) . We lift and extend the path of flat connections A t to Z ,denoted by ˜ A t , so that ˜ A is the product connection. The function f : V → S is also liftedand extended to a function τ : Z → R as before. Taubes’ criterion [Tau87, Lemma 4.3] impliesthat Q ˜ A i ,δ ( Z ) = ( − d ∗ ˜ A i − δι dτ , d +˜ A i − δ ( dτ ∧ − ) + ) : L ( T ∗ Z ⊗ C ) → L ( Z, C ) ⊕ L (Λ + T ∗ Z ⊗ C ) is Fredholm for i = 0 , . Given a perturbation π ∈ P , we can equally lift and extend to one,say ˜ π , on Z . Since π ∈ L k and k ≥ , the Sobolev multiplication gives us a bounded map ˜ π : L ( T ∗ Z ⊗ C ) → L (Λ + T ∗ Z ⊗ C ) which is not necessarily compact. The operator norm of ˜ π is bounded by k π k L k + k ˜ π | M k L k , which can be chosen as small as we want allowing varying π generically. Let’s denote by Q ˜ A i ,δ ( Z, ˜ π ) the operator obtained by perturbing the secondcomponent of Q ˜ A i ,δ ( Z ) using the lift ˜ π of π . Now applying [MRS11, Theorem 7.3], we get(3.7) ind Q ˜ A ,δ ( Z, ˜ π ) − ind Q ˜ A ,δ ( Z, ˜ π ) = e Sf( Q A t ( V, π )) . On the other hand, by choosing lifts ˜ π of small operator norm, we have(3.8) ind Q ˜ A i ,δ ( Z, ˜ π ) = ind Q ˜ A i ,δ ( Z ) . Since δ ∈ (0 , ǫ ) , we can apply Theorem 2.3 to get − ˜ ρ ϕ ( V, Q ) = ρ ( V, Q ) − ρ ( V, Q ϕ ) = e Sf( Q A t ( V, π )) . (cid:3) Remark 3.11.
A priori, the choice of perturbations π depends on the choice of δ . However, ifwe only allow δ to vary within a compact subset of (0 , δ ) , the perturbations π can be chosento work for all δ in this range using the argument in Lemma 3.2. Thus the corresponding QUIVARIANT TORUS SIGNATURE AND PERIODIC RHO INVARIANT 17 spectral flow does not depend on the choices of small π and δ in this range. When δ = 0 , twoissues might arise. In the first place, defining periodic spectral flow requires perturbing the firstcomponent of the ASD DeRham operator Q due to the appearance of the product connection.After adding such perturbations, the periodic spectral flow might depend on the choice of smallperturbations since we are now looking at spectral points on the unit circle for each connectionon the path. 4. Singular Furuta-Ohta Invariants
In this section, we first recall the definition of the singular Furuta-Ohta invariant introducedby Echeverria [Ech19] by counting singular instantons over homology S × S . Then we derivean equivalent interpretation using the structure theorem of the moduli space of degree zeroinstantons over a homology D × T deduced by the author [Ma20].4.1. The Definition.
We briefly recall the set-up of singular instantons in [KM93]. Let X be a smooth connectedclosed -manifold with an embedded surface Σ . Let E be an SU (2) -bundle over X with areduction E | ν (Σ) = L ⊕ L ∗ in a neighborhood ν (Σ) of the surface. Such a bundle is characterizedby two topological quantities:(4.1) m = c ( E )[ X ] and l = − c ( L )[Σ] , which are referred to as the instanton number and monopole number respectively. We can regard ν (Σ) as a disk bundle over Σ . Let η ∈ Ω ( ν (Σ) \ Σ , R ) be the angular form on the punctured diskbundle. Pick a smooth unitary connection A on E as a reference connection. To a holonomyparameter α ∈ (0 , , we assign a model connection:(4.2) A α = A + iβ (cid:18) α − α (cid:19) η, where β is a cut-off function on the radial direction of ν (Σ) that equals near Σ . We think of A α as a unitary connection on E | X \ Σ .So far the notion of metric has not entered the picture. Let ν be a positive integer. Wedenote by g ν a metric on X that is smooth on X \ Σ and singular along Σ with cone-angle π/ν .Alternatively we can think of g ν as an orbifold metric after turning X into an orbifold whosesingular set is Σ with isotropy group Z /ν . Let’s write g E for the adjoint bundle associated to E . We denote by ˇ L k ( X, Λ ∗ X ⊗ g E ) the space of g E -valued forms completed by the L k Sobolevnorm associated to the Levi-Civita connection of g ν and A α . The key analytic input in thetheory of singular instantons is the following result proved by Kronheimer-Mrowka [KM93]. Proposition 4.1 ([KM93, Proposition 4.17]) . Given any compact interval I ⊂ (0 , / andinteger k > , there exists ν > such that for all ν ≥ ν and k ≤ k the operator Q A α = − d ∗ A α ⊕ d + A α : ˇ L k ( X, T ∗ X ⊗ g E ) −→ ˇ L k − ( X, (Λ ⊕ Λ + ) T ∗ X ⊗ g E ) is Fredholm, L -self-adjoint, and satisfies the Fredholm alternative. With Proposition 4.1 in hand, we define the configuration space of singular connections tobe A αk ( X, Σ , E ) := A α + ˇ L k ( X, T ∗ X ⊗ g E ) . Denote by G ( E ) the ˇ L k +1 -gauge group. Then we have the moduli space of α -singular instantons: M αm,l ( X, Σ) := (cid:8) A ∈ A αk ( X, Σ , E ) : F + A = 0 (cid:9) / G ( E ) . A priori, the moduli space depends on the cone-angle parameter ν . We say a singular connection A ∈ A α ( X, Σ , E ) is reducible if the splitting L ⊕ L ∗ over ν (Σ) can be extended to all X andrespects A . Otherwise the A is called irreducible.Now we restrict our attention to the following particular case. The -manifold X is a ho-mology S × S , and the surface is an essentially embedded torus denoted by T . We let E bethe trivial C -bundle over X with a trivial reduction near T , in which case both the instantonand monopole numbers vanish. Due to the Chern-Weil formula [KM93, Proposition 5.7], ananti-self-dual α -singular connection A on E is flat. Taking the holonomy we get an element inthe character variety(4.3) χ α ( X, T ) := (cid:8) ϕ : π ( X \ Σ) → SU (2) : ϕ ( µ T ) ∼ e πiα (cid:9) / Ad , where µ T is a meridian of T . In fact, this assignment is a one-to-one correspondence between M α ( X, T ) and χ α ( X, T ) . We shall denote χ α, red ( X, T ) for the subspace of reducible represen-tations. We identify U (1) as a standard maximal torus of SU (2) by(4.4) e iθ (cid:18) e iθ e − iθ (cid:19) . Given a reducible representation [ ϕ ] ∈ χ α, red ( X, T ) , we choose the representative ϕ : π ( X \ Σ) → U (1) to satisfy ϕ ( µ T ) = e − πiα , which is consistent with the holonomy of A α on the first factorof the splitting E = L ⊕ L ∗ . To define the singular Furuta-Ohta invariant, we need to introducethe following non-degeneracy condition on χ α ( X, T ) . Definition 4.2.
We say a reducible α -representation [ ϕ ] ∈ χ α, red ( X, T ) is non-degenerate if H ( X \T ; C ϕ ) = 0 . We say χ α, red ( X, T ) is non-degenerate if every reducible α -representationis non-degenerate. Remark 4.3.
As pointed out by Echeverria [Ech19], the non-degeneracy condition is equiv-alent to the non-vanishing of the Alexander polynomial of T evaluating on the correspondingrepresentation.The singular Furuta-Ohta invariant introduced by Echeverria [Ech19] is defined as follows.Let ( X, T ) be a pair as above. For each α ∈ (0 , / with χ α, red ( X, T ) non-degenerate, onecan perturb the moduli space M α ( X, T ) to get a compact -dimensional irreducible part, say M α, ∗ σ ( X, T ) , where σ stands for the perturbation. A choice of the generator H ( X, Z ) providesus a homology orientation. Then the singular Furuta-Ohta invariant is defined to be the signedcount:(4.5) λ F O ( X, T , α ) := M α, ∗ σ ( X, T ) , QUIVARIANT TORUS SIGNATURE AND PERIODIC RHO INVARIANT 19 which is independent of the choice of generic perturbations. Note that for each α ∈ (0 , / satisfying the non-degeneracy condition we need an orbifold metric g ν to apply the Fredholmpackage of instanton theory. The moduli space M α ( X, T ) thus depends on the choice of thecone-angle π/ν . We will show, later in this section, that the singular Furuta-Ohta invariantis independent of such a choice, which provides a positive answer to this dependence questionraised by Echeverria [Ech19].4.2. Instantons over Manifolds with Cylindrical End.
We review the structure theorem of degree zero instantons over a homology D × T with acylindrical end modeled on [0 , ∞ ) × T deduced by the author in [Ma20].Let M o = M ∪ [0 , ∞ ) × T be a -manifold with cylindrical end, where M is a compact -manifold with the same integral homology of D × T . We fix a metric g on M o of the form g | [0 , ∞ ) × T = dt + h , where h is a flat metric on T . We let E be the trivial C -bundle over M o .Let A k be the space of L k,loc SU (2) -connections on E , and G k +1 the space of L k +1 ,loc gaugetransformations. The action and energy of a connection A ∈ A k are defined respectively to be(4.6) κ ( A ) := 18 π Z M o tr( F A ∧ F A ) E ( A ) := Z M o | F A | . The unperturbed moduli space of degree zero instantons over Z is given by(4.7) M ( M o ) := { A ∈ A k : F + A = 0 , κ ( A ) = 0 , E ( A ) < ∞} / G k +1 . Let µ > be a weight. As in [Ma20, Subsection 2.2], one can consider a Banach space P µ ( M o ) of holonomy perturbations on A k which stand as G k +1 -equivariant maps(4.8) σ : A k −→ L k,µ ( Z, Λ + T ∗ Z ⊗ su (2)) . From its construction, we see that the differential Dσ | A ∈ P µ , the perturbation space consideredin (3.6), for each A ∈ A k . The σ -perturbed moduli space is denoted by M σ ( M o ) . Let χ ( T ) :=Hom( π ( T ) , SU (2)) / Ad be the SU (2) -character variety of the -torus. In [Ma20, Theorem1.7] the author deduced an asymptotic map(4.9) ∂ + : M σ ( M o ) −→ χ ( T ) by showing the limit [ A | { t }× T ] exists as t → ∞ for all [ A ] ∈ M σ ( M o ) and σ ∈ P µ ( M o ) .Inspired by the treatment in [Her97], it would be easier to work with the double cover of χ ( T ) for our purpose. We let ˜ χ ( T ) := Hom( π ( T ) , U (1)) the U (1) -character variety of T ,which is a double cover of χ ( T ) branched along the eight central representations. Then ˜ χ ( T ) is equivalent to the space of flat U (1) -connections on the trivial line bundle of T modulo U (1) -gauge transformations. We note that the identification (4.4) of U (1) in SU (2) gives us anidentification i R ⊕ C ≃ su (2) by(4.10) ( v, z ) (cid:18) v z − ¯ z − v (cid:19) . We consider the restricted configuration space(4.11) ˜ A k := n A ∈ A k : ( A | { t × T − d ) C → in L k − / ( Y ) o , where ( A | { t × Y } − d ) C means the projection to the C component with respect to the decomposi-tion (4.10). Then ˜ A k consists of connections that are asymptotically abelian. We also considerthe restricted gauge group(4.12) ˜ G k +1 := n u ∈ G k +1 : ( du t · u − t ) C → in L k − / ( Y ) o , where u t = u | { t }× Y . Then for each holonomy perturbation, we get a perturbed moduli space(4.13) f M σ ( M o ) := { A ∈ ˜ A k : F + A = σ ( A ) , κ ( A ) = 0 , E ( A ) < ∞} / ˜ G k +1 . The corresponding asymptotic map is denoted by ˜ ∂ + : f M σ ( M o ) −→ ˜ χ ( T ) . Lemma 4.4. f M σ ( M o ) is a double cover of M σ ( M o ) branched along central instantons.Proof. Since χ ( T ) consists of abelian flat connections, each instanton [ A ] ∈ M σ ( M o ) admitsrepresentatives in ˜ A k . Since ˜ G k +1 is a subgroup of G k +1 , we get a surjective map p : f M σ ( M o ) →M σ ( M o ) .Let’s denote by G ′ k +1 ⊂ G k +1 the space of gauge transformations that preserve ˜ A k . We claimthat ˜ G k +1 is an index two subgroup of G ′ k +1 . Note that elements in ˜ G k +1 take value in U (1) asymptotically along the cylindrical end. To preserve ˜ A k , elements in G ′ k +1 take value in thenormalizer of U (1) asymptotically. The normalizer of U (1) is P in (2) = U (1) ∪ j · U (1) ⊂ SU (2) .Thus for each element in G ′ k +1 takes the form of either u or j · u for some u ∈ ˜ G k +1 .When [ A ] ∈ M σ ( M o ) is non-central, the stabilizer of A is either Z / or U (1) . Thus [ j · A ] = [ A ] in f M σ ( M o ) . When [ A ] ∈ M σ ( M o ) is central, j · A = A . This finishes the proof. (cid:3) With the help of Lemma 4.4 the structural theorem for M σ ( M o ) in [Ma20] can be adaptedto the double cover f M σ ( M o ) directly. Theorem 4.5 ([Ma20, Theorem 1.9]) . With respect to generic small holonomy perturbations σ ∈ P µ ( M o ) , the moduli space f M σ ( M o ) is a compact smooth oriented stratified space of thefollowing structure.(a) The reducible locus f M red σ ( M o ) is diffeomorphic to a -torus.(b) The irreducible locus f M ∗ σ ( M o ) is a smooth oriented -manifold of finitely many compo-nents, each of which is either diffeomorphic to S or (0 , .(c) The ends of the arcs in f M ∗ σ ( M o ) lie in f M red σ ( M o ) but stay away from the central in-stantons. Near each end [ A ] ∈ f M red σ ( M o ) , the moduli space f M σ ( M o ) is modeled on aneighborhood of in o − (0) ⊂ R ⊕ R + , where o : R ⊕ R + −→ R ( x , x , r ) ( x + ix ) · r is the Kuranishi obstruction map.(d) The asymptotic map ˜ ∂ + : f M ∗ σ ( M o ) → ˜ χ ( T ) is C transverse to a given submanifold.(e) The image of the irreducible locus f M ∗ σ ( M o ) under the asymptotic map ˜ ∂ + misses centralflat connections in ˜ χ ( T ) . QUIVARIANT TORUS SIGNATURE AND PERIODIC RHO INVARIANT 21
The end points [ A ] ∈ f M red σ ( M o ) appeared in (c) of Theorem 4.5 is referred to as bifurcationpoints. In the following, we see that these points are characterized by looking at the cohomologyof the deformation complex. The deformation complex at [ A ] is given by( E SU (2) A,δ,σ ) ˆ L k +1 ,δ ( M o , su (2)) − d A −−−→ ˆ L k,δ ( T ∗ M o ⊗ su (2)) d + A,σ −−−→ L k − ,δ (Λ + T ∗ M o ⊗ su (2)) , where ˆ L k,δ ( M o , su (2)) means su (2) -valued functions on M o that are asymptotically to i R -valuedfunctions with respect to the norm L k +1 ,δ , and similarly for ˆ L k,δ ( T ∗ M o ⊗ su (2)) . Since [ A ] isreducible, we write A = A l ⊕ A ∗ l with respect to the splitting E = C ⊕ C , and A † = A ⊗ l . Then( E SU (2) A,δ,σ ) splits into the direct sum of( E U (1) δ,σ ) ˆ L k +1 ,δ ( M o , i R ) − d −−→ ˆ L k,δ ( T ∗ M o ⊗ i R ) d + σ −−→ L k − ,δ (Λ + T ∗ M o ⊗ i R ) and( E A † ,δ ) L k +1 ,δ ( M o , C ) − d A † −−−→ L k,δ ( T ∗ M o ⊗ C ) d + A † ,σ −−−→ L k − ,δ (Λ + T ∗ M o ⊗ C ) . Then [ A ] is a bifurcation point if and only if(4.14) H ( E A † ,δ ( M o )) = C . It was proved in [Ma20] that there are only finite many bifurcation points with respect to smallgeneric perturbations in P µ . Moreover all of them are away from the central instantons in f M red ( M o ) . For any points in f M red ( M o ) other than the central ones, the firs cohomology of( E A † ,δ ) vanishes. Let’s denote the set of bifurcation points in f M red σ ( M o ) by f Bf( M o , σ ) .The orientability of the moduli space f M σ ( M o ) can be worked out the same as that in[Her94, Section 9]. In particular the involution on the double covers f M σ ( M o ) and ˜ χ ( T ) arenot orientation-preserving. Since the bifurcation points are the ends of the oriented irreduciblelocus f M ∗ σ ( M o ) , one can assign a sign given by the boundary orientation to each bifurcationpoints. This gives us a well-defined count f Bf( M o , σ ) , which will play an important role in theproof of Theorem 1.1. In practice the sign is determined as follows.When [ A ] is not a central instanton, one can choose δ > so that ( E SU (2) A,δ ) is a Fredholmcomplex. Since we are using holonomy perturbations which does not perturb central instantons,we may fix a small neighborhood O of the central flat connections in ˜ χ ( T ) so that f M red σ ( M o , O c ) := ˜ ∂ − ( O c ) ∩ f M red σ ( M o ) contains no central instantons, where O c = ˜ χ ( T ) \O is the complement of O . Then one canchoose small δ > so that ( E SU (2) A,δ ) is Fredholm for all [ A ] ∈ f M red σ ( M o , O c ) . Since ( E A † ,δ ) iscomplex, we get a canonical trivialization on its determinant line bundle. The trivialization of ( E U (1) δ ) is given by the homology orientation of M o . Thus we get a preferred trivialization ofthe determinant line det E SU (2) A,δ over f M red ( M o , O c ) . Now let [ A ] ∈ f Bf( M o , σ ) be a bifurcation point. Let’s write H iA for the i -th cohomology of ( E SU (2) A,δ,σ ) . Then H A = i R , H A = i R ⊕ i R ⊕ C , H A = C . We fix an ordered real basis for these spaces that agree with the trivialization of the determinantline above. Since
Stab A = U (1) , then U (1) acts on the complex part of H A and H A with weight . One applies the Kuranishi argument to obtain a U (1) -equivariant map q A : H A → H A so thatthe moduli space near [ A ] is modeled on the U (1) -quotient of the zero set of the obstructionmap(4.15) o A ( a ) = Π A (cid:16) F + A + a + q A ( a ) − σ ( A + a + q A ( a )) (cid:17) , where Π A is the L δ -projection onto H A . As proved in [Ma20, Proposition 5.11], the lowest orderterm of o A takes the form f ( x , x ) · z with f : i R ⊕ i R → C a function with non-vanishing firstorder term sending (0 , to . Since the irreducible locus near [ A ] is given by the U (1) -quotientof { (0 , , z ) ∈ o − A (0) } , so we assign +1 to [ A ] if f is orientation-preserving and − if f isorientation-reversing. Intuitively, assigning +1 means the orientation of the irreducible locus ispointing away from the reducible locus, and − for the converse.4.3. A Reformulation of λ F O . We are going to deduce another formulation of the singular Furuta-Ohta invariant using agluing argument based on Theorem 4.5.Let X be an integral homology S × S with an essentially embedded torus T . We also fixa primitive class X ∈ H ( X ; Z ) serving the role of homology orientation. Let M = X \ ν ( T ) be the complement of a tubular neighborhood of T . From the Mayer-Vietoris sequence, we seethat M is a homology D × T with ∂M = T . We first choose a basis of curves { µ, λ, γ } on ∂ν ( T ) = − ∂M as follows: • [ µ ] generates ker H ( ∂ν ( T ); Z ) → H ( ν ( T ); Z ) ; • [ λ ] generates ker H ( ∂ν ( T ); Z ) → H ( M ; Z ) ; • X · [ γ ] = 1 .Up to isotopy and orientation-reversing, µ and λ are uniquely determined. We will not specifythe isotopy class of γ , but simply make a choice.To a flat U (1) -connection B = d + b on T , we assign coordinates x ( B ) = 12 πi Z µ b, y ( B ) = 12 πi Z λ b, z ( B ) = 12 πi Z γ b, which take values in R / Z . The holonomies of B around µ, λ, γ are given respectively by Hol µ B = e − πix ( B ) , Hol λ B = e − πiy ( B ) , Hol γ B = e − πiz ( B ) . Let C T = { ( x, y, z ) : x, y, z ∈ [ − / , / } be the fundamental cube so that ˜ χ ( T ) is identifiedwith the quotient of C T by identifying opposite faces of the cube. The SU (2) -character varietyis identified as χ ( T ) = ˜ χ ( T ) / ∼ under the antipodal relation xxx ∼ − xxx . Given x ∈ [ − / , / ,we get a -torus e T x ⊂ ˜ χ ( T ) given by the quotient of the plane in C T with fixed x -coordinate x . We write T x = e T x / ∼ for its quotient in χ ( T ) . We note that when x ∈ {− / , / } , QUIVARIANT TORUS SIGNATURE AND PERIODIC RHO INVARIANT 23 e T x is a double cover of T x branched along central connections. When x ∈ ( − / , / , e T x is diffeomorphic to T x as a -torus. Lemma 4.6.
We continue with the notations as above. Let α ∈ (0 , / be a holonomy parame-ter satisfying that χ α, red ( X, T ) is non-degenerate. Then with respect to generic small holonomyperturbations σ , we have λ F O ( X, T , α ) = M ∗ σ ( M o ) ∩ ∂ − ( T α ) . Proof.
Let’s write N o = ( −∞ , × T ∪ ν ( T ) . Given α ∈ (0 , / , we put a metric g ν on X with cone angle π/ν along T satisfying Proposition 4.1 and cylindrical near ∂ν ( T ) . Since weare working with instantons of topological action , the neck-stretching process causes no lostof action. Recall in Theorem 4.5, the asymptotic map ∂ + : M ∗ σ ( M o ) → χ ( T ) can be made tobe transverse to any given submanifold in χ ( T ) and misses the central flat connections. Thenwhen the length T of the "neck" is long enough, each irreducible α -singular instanton [ A T ] on ( X T , T ) is glued by one irreducible instanton [ A M ] ∈ M ∗ σ ( M o ) and one reducible α -singularinstanton on [ A N ] ∈ M α, red ( N o , T ) whose asymptotic values agree in χ ( T ) . Note that N o issimply the completion of the product D × T . The unperturbed moduli space of reducible α -singular flat connections is smoothly cut out due to the vanishing of b + . So we don’t needto perturb the moduli space further to achieve transversality. Thus the singular Furuta-Ohtainvariant can be expressed as the count of the fiber product: λ F O ( X, T , α ) = M ∗ σ ( M o ) × ( ∂ + ,∂ − ) M α, red ( N o , T ) . Now it remains to identify ∂ + ( M α, red ( N o , T )) with T α ⊂ χ ( T ) . Since M α, red ( N o , T ) consistsof flat connections whose holonomy around the meridian of T is e − πiα , the image under theasymptotic map is given by restricting to ∂ν ( T ) which is T α by construction. (cid:3) Remark 4.7.
This formulation of the singular Furuta-Ohta invariant implies that it’s inde-pendent of the choice of cone angles π/ν of metrics along the singular surface T .In the end, we remark on the definition of the usual Furuta-Ohta invariant for an integralhomology S × S [FO93]. Just like the non-degeneracy condition for defining singular Furuta-Ohta invariants, the corresponding non-degeneracy condition for the non-singular case is thefollowing:(4.16) H ( X ; C ϕ ) = 0 for all non-trivial representation ϕ : π ( X ) → U (1) . When this non-degeneracy condition is satisfied, the Furuta-Ohta invariant for X together witha fixed primitive class X ∈ H ( X ; Z ) is defined to be(4.17) λ F O ( X ) := 14 M ∗ σ ( X ) a quarter of the signed count of irreducible instantons over the trivial bundle of X under smallgeneric perturbations. By the same neck-stretching argument as in Lemma 4.6, one can showthat with respect to generic small perturbations σ (4.18) λ F O ( X ) = 14 M ∗ σ ( M o ) ∩ ∂ − ( T ) = 18 f M ∗ σ ( M o ) ∩ ˜ ∂ − ( e T ) , since ˜ T is a double cover of T and the image of f M ∗ σ ( M o ) under the asymptotic map missesthe branching points.4.4. A Sugery Formula.
In this section we prove the surgery formula Theorem 1.5 using the reformulation of thesingular Furuta-Ohta invariant. Given the pair ( X, T ) of an essentially embedded torus in ahomology S × S , the /q -surgery of X along T is given by X /q ( T ) = X \ ν ( T ) ∪ ψ q D × T , where ψ q : ∂D × T → ∂ν ( T ) is given by the matrix ψ q = q under the basis of curves { µ, λ, γ } chosen as in subsection 4.3. We are going to prove thefollowing surgery formula. Proposition 4.8.
Let α ∈ (0 , / be a non-degenerate parameter for the pairs ( X, T ) , ( X ( T ) , T ) ,and ( X ( T ) , T ) . Then λ F O ( X ( T ) , T , α ) = λ F O ( X, T , α ) + λ F O ( X ( T ) , T , α ) . Note that with respect to the framing of T q in X /q ( T ) , performing -surgery and -sugerygives us X / ( q +1) ( T ) and X ( T ) respectively. Thus Theorem 1.5 is proved by applying Proposition 4.8repetitively. Proof of Proposition 4.8.
For y ∈ ( − / , / , we denote by ˜ S y the -torus in ˜ χ ( T ) withfixed y -coordinate y . Then the neck-stretching argument tells us that(4.19) λ F O ( X ( T ) , T , α ) = f M ∗ σ ( M o ) ∩ ˜ ∂ − ( ˜ S α ) . We denote by ˜ P ⊂ ˜ χ ( T ) the -torus given by { x + y = α } . Then(4.20) λ F O ( X ( T ) , T , α ) = f M ∗ σ ( M o ) ∩ ˜ ∂ − ( ˜ P α ) . Denote by J the involution on ˜ χ ( T ) given by J · xxx = − xxx . Since the asymptotic map intertwineswith the involutions, we write J for the involution on f M ∗ σ ( M o ) as well. Then we know(4.21) f M ∗ σ ( M o ) ∩ ˜ ∂ − ( ˜ S α ) = J f M ∗ σ ( M o ) ∩ ˜ ∂ − ( J ˜ S α ) = f M ∗ σ ( M o ) ∩ ˜ ∂ − ( J ˜ S α ) since J reverse the orientation for both f M ∗ σ ( M o ) and ˜ ∂ − ( ˜ S α ) , and J ˜ S α ∩ ˜ S α = ∅ . Note thatthe union(4.22) ˜ T α ∪ J ˜ T α [ ˜ S α ∪ J ˜ S α [ − ˜ P α ∪ − J ˜ P α is null-homologous in ˜ χ ( T ) \ ˜ C , where ˜ C is the set of the eight central representations. Thus theunion (4.22) bounds a -complex in ˜ χ ( T ) \ ˜ C , say ˜ V . We can use Theorem 4.5 to pick a generic QUIVARIANT TORUS SIGNATURE AND PERIODIC RHO INVARIANT 25 small perturbation so that ˜ ∂ + | ˜ M ∗ is transverse to ˜ V . Then f M ∗ σ ( M o ) ∩ ˜ ∂ − ( ˜ V ) is an oriented -manifold with boundary, and the surgery formula follows from the observation that ∂ (cid:16) f M ∗ σ ( M o ) ∩ ˜ ∂ − ( ˜ V ) (cid:17) = 0 . (cid:3) Equivariant Torus Signature
In this section, we will prove the main results concerned in this article. Let’s recap thesettings to start with. Let ( X, T ) be the pair consisting of an essentially embedded torus in anintegral homology S × S . We decompose X = M ∪ ν ( T ) into a homology D × T with atubular neighborhood of T . We fix a primitive class X ∈ H ( X ; Z ) . A basis of curves { µ, λ, γ } are chosen as in subsection 4.3 on ∂ν ( T ) = − ∂M , where the isotopy classes of µ and λ aredetermined. Such a basis provides us with a framing ν ( T ) ≃ D × T . Then we can perform -surgery along T to get V = M ∪ ψ D × T with the gluing map ψ : ∂D × T → ∂M specifiedby ψ = . Note that the gluing map preserves the third curve γ . So the surgery is well-defined despiteof the ambiguity of choosing γ . Either by Mayer-Vietoris sequence or a geometric argumentgiven in [Rub20, Section 3.1], X gives rise to a primitive class V ∈ H ( V ; Z ) . In the samearticle, Ruberman also showed that V is a cohomology T × S endowed with a degree- mapto T × S . We pick a smooth function f : X → S representing X so that its restriction f | ν ( T ) is the projection to the S -factor corresponding to γ . In this way, f is invariant underthe gluing map ψ , thus gives rise to a map, still denoted by f , on the -surgered manifold V representing V .We will abuse the notations µ, λ, γ for corresponding curves in either X or V . We also write µ ∨ , λ ∨ , γ ∨ for the corresponding cohomology classes in various manifolds. For instance, γ ∨ couldbe either X or V depending on the context.5.1. Proof of Theorem 1.1.
To a holonomy parameter α ∈ (0 , / , we assign a representation ϕ α : π ( V ) → U (1) by(5.1) ϕ α ( µ ) = e − πiα and ϕ α ( γ ) = 1 . Lemma 5.1. χ α, red ( X, T ) is non-degenerate if and only if H ∗ ( E ϕ α ,z ( V )) = 0 for all | z | = 1 .Proof. Let [ ϕ ] ∈ χ α, red ( X, T ) be an α -reducible representation. Since H ( X \T ) and H ( V ) areboth freely generated by [ µ ] and [ γ ] . We can regard ϕ as a representation from π ( V ) to U (1) .By our convention ϕ ( µ ) = e − πiα . Let B α , B ϕ be flat connections corresponding to ϕ and ϕ α respectively. From their holonomies we can tell B α − B ϕ = ln z · df for some | z | = 1 . Since H ( D × T , C ϕ ) = 0 for all [ ϕ ] ∈ χ α, red ( X, T ) , the Mayer-Vietoris sequence tells us that H ( X \T , C ϕ ) ≃ H ( V, C ϕ ) = H ( E ϕ α ,z ) . As [ ϕ ] ranges over χ α, red ( X, T ) , z ranges over the unit circle. This concludes the proof combiningwith the fact that H ( E ϕ α,z ) = 0 and χ ( E ϕ α ,z ) = 0 . (cid:3) Now let α be a holonomy parameter satisfying the non-degenerate condition in Definition 4.2.We further assume that V satisfies the non-degenerate condition (4.16). Then Lemma 4.6 and(4.18) tells us that(5.2) λ F O ( X, T , α ) − λ F O ( X ) = f M ∗ σ ( M o ) ∩ ∂ − ( e T α ) − f M ∗ σ ( M o ) ∩ ˜ ∂ − ( e T ) . Let e P [0 ,α ] = { ( x, y, z ) ∈ C T : x ∈ [0 , α ] } / ∼ be the product torus in e χ ( T ) with boundary ∂ e P [0 ,α ] = − e T ∪ e T α . It follows from Theorem 4.5 that the closure of f M ∗ σ ( M o ) ∩ ∂ − ( e P [0 ,α ] ) inthe moduli space f M σ ( M o ) is an oriented compact manifold whose boundary consists of f M ∗ σ ( M o ) ∩ ∂ − ( e T α ) [ − f M ∗ σ ( M o ) ∩ ˜ ∂ − ( e T ) [ f Bf( M o , σ ) ∩ ∂ − ( e P [0 ,α ] ) , where f Bf( M o , σ ) is the set of bifurcation points in f M red σ ( M o ) . Thus we conclude that(5.3) λ F O ( X, T , α ) − λ F O ( X ) = − f Bf( M o , σ ) ∩ ∂ − ( e P [0 ,α ] ) . Let ( A t ) be a path of flat SU (2) -connections on M o defined as follows. A is the productconnection and A the flat connection whose holonomies around µ and γ are respectively e − πiα and . Let A t = A + t ( A − A ) be the path from A to A . With respect to the splitting E = C ⊕ C , we write A t = A l ,t ⊕ A ∗ l ,t and A l ,t = d + a l ,t with a l ,t ∈ Ω ( M o , i R ) . Let’s denote by A † t = d + 2 a l ,t the path of flat connections on the trivial bundle C , and by B † t = A † t | { }× T therestriction to a slice of the end. Then the coordinates of B † t in ˜ χ ( T ) are given respectively by x ( B † t ) = 2 tα, y ( B † t ) = 1 , z ( B † t ) = 1 . Lemma 5.2.
There exists δ > such that for all δ ∈ (0 , δ ) , the path ( A † t ) consists of δ -regularly asymptotically flat connections in the sense of Definition 3.8.Proof. Since A † t is flat, the first condition (a) in Definition 3.8 is automatic. To check the secondcondition can be satisfied, let’s write B † t,z = B † t − ln z · df . Note the operator L B † t,z decomposeswith respect to Ω ( T , C ) ⊕ im d B † t,z ⊕ ker d ∗ B † t,z into the sum − d ∗ B † t,z − d B † t,z ! ⊕ ⋆d B † t,z | ker d ∗ B † t,z . When B is the product connection, we know L B is self-adjoint, thus ind L B = 0 . Since thedifference L B † t,z − L B is a zero-th order differential operator, we know ind L B † t,z = 0 as well.Thus it suffices to show that ker L B † t,z = 0 for all ( t, z ) ∈ [0 , × S δ . Note that when | z | 6 = 1 , B † t,z cannot be the product connection. So the kernel can only survive in the third component ker d ∗ B † t,z , which corresponds to B † t,z -harmonic -forms on T . Such harmonic -forms are zerounless B † t,z is the product connection. This verifies the second condition (b) of Definition 3.8. QUIVARIANT TORUS SIGNATURE AND PERIODIC RHO INVARIANT 27
To verify the third condition (c), Note that for ( t, z ) ∈ [0 , × S δ , Q A † t,z is always Fredholm,thus its index is unaltered with respect to varying ( t, z ) here. In particular, by letting t = 0 and z = e δ , the vanishing of the index follows from the Atiyah-Patodi-Singer theory [APS75a]and the fact that M o is a homology D × T . (cid:3) With Lemma 5.2, the periodic spectral flow of this path ( A † t ) is well-defined with respect toa generic small perturbations π ∈ P µ as in subsection 3.1. Proposition 5.3.
Let ( A † t ) be the path of flat connections on M o as above. Then we have f Bf( M o , σ ) ∩ ∂ − ( e P [0 ,α ] ) = e Sf( Q A † t ( M o , π )) with respect to generic small perturbations σ ∈ P µ and π ∈ P µ .Proof. Let σ ∈ P µ be a generic small holonomy perturbation satisfying Theorem 4.5. As shownin [Ma20, Proposition 5.6], one can find a path σ s , s ∈ [0 , , of holonomy perturbations in P µ from to σ such that the parametrized moduli space Z I := [ s ∈ [0 , { s } × (cid:16) f M red σ s ( M o ) ∩ ˜ ∂ − ( e P [0 ,α ] ) (cid:17) is a product cobordism. We also write Z s := { s }× (cid:16) f M red σ s ( M o ) ∩ ˜ ∂ − ( e P [0 ,α ] ) (cid:17) . To each [ A ] ∈ Z I ,we assign a connection [ A † ] on the trivial bundle C . Such an assignment gives rise to a familyof connections Z † I on C . For each [ A † ] ∈ Z † I , the ASD DeRham operator Q A † : L ,δ → L δ isFredholm of index zero. Thus we can apply the transversality argument in Lemma 3.2 to getgeneric path of small perturbations π s ∈ P µ , with π = 0 , so that the set S † I := n [ A † ] ∈ Z † s : ker Q A † ,σ s ,π s = 0 o ⊂ Z † I is a compact submanifold of codimension- . Moreover for each [ A † ] ∈ S † I , ker Q A † ,σ s ,π s = C .Everything above can be made oriented once we choose a trivialization of the index bundleof the SU (2) ASD DeRham operator over the configuration space of SU (2) -connections over M o exponentially asymptotic to flat connections as in the usual Yang-Mills theory [Don87].The exponential convergence of instantons on f M red σ s ( M o ) was derived in [Ma20]. Thus we get awell-defined count on the compact -dimensional manifold ∂ S † I = S † ∪ −S † .Converting from S † I to S I as SU (2) -connections, we see that S is precisely the set of bifur-cation points f Bf( M o , σ ) ∩ ∂ − ( e P [0 ,α ] ) . Let’s write π = π . Thus we get f Bf( M o , σ ) ∩ ∂ − ( e P [0 ,α ] ) = S † = S † . So all that left is to identify S † with the periodic spectral flow e Sf( Q A † t ( M o , π )) . Note that thespectral flow counts points with non-trivial kernel on the cylinder [0 , × S δ for the operators Q A † t,z ,π : L → L . Using the identification L k,δ = e − δf L k , we see it’s equivalent to countpoints with non-trivial kernel on the cylinder [0 , × S for Q A † t,z ,π : L ,δ → L δ , which preciselycorrespond to points in S † . Now we show that the sign of each point in S † agrees with that in the definition of spectralflow. Since along each component of S † I the dimension of ker Q is constant, the orientationtransport is canonical. Since we only allow small perturbations, the orientation transport iscanonical as well along two perturbations. So we may assume [ A ] ∈ M red ( M o ) is a bifurcationpoint which gives a regular δ -RAF connection [ A † ] ∈ S † in the first place. Under the notationof subsection 4.2, we have H A = i R , H A = i R ⊕ i R ⊕ C , H A = C , where i R ⊕ i R = H ( M ; i R ) in H A . Let { idg, idf } be an ordered basis for H ( M ; i R ) where dg represents µ ∨ , and df represents γ ∨ as before. Let A † x = A † + x · idg − ln z ( x ) · df , x ∈ ( − ǫ, ǫ ) , bea local spectral curve in [0 , × C ∗ near A † . Working near [0 , × S , we may write ln z ( x ) = r ( x ) + is ( x ) with r (0) = 0 , s (0) = 0 and ˙ r (0) = 0 . The contribution of A † x in the periodicspectral flow is given by the sign of ˙ r (0) . We need to show this is the sign at [ A ] .Since ker Q A † x = C for all x ∈ ( − ǫ, ǫ ) , we can pick a path of non-zero -forms φ † x ∈ ker Q A † x .Let A x be the path of connections on the trivial C -bundle given by A † x and φ x = (0 , φ † x ) the su (2) -valued -form associated to φ † x . We let { φ , iφ } be an ordered basis for the C -componentof H A . Denote by Π A : L k − ,δ (Λ + ⊗ su (2)) → H A the L δ -projection. Then the second orderterm of o A in (4.15), with respect to the ordered basis { idg, idf, φ , iφ } of H A , takes the form(5.4) D o A : ( x , x , x , x ) · Π A (( x idg + x idf ) ∧ ( x φ + x iφ )) + . Note that(5.5) d + A x φ x = d + A φ x + x · ( dg ∧ iφ x ) + − ln z ( x ) · ( df ∧ φ x ) + = 0 Differentiating (5.5) at x = 0 gives us d + A ˙ φ = − ( idg ∧ φ ) + + ˙ r (0)( df ∧ φ ) + + ˙ s (0)( df ∧ iφ ) + , which implies that(5.6) Π A ( idg ∧ φ ) + = ˙ r (0)Π A ( df ∧ φ ) + + ˙ s (0)Π A ( df ∧ iφ ) + . We claim that { Π A ( df ∧ φ ) + , Π A ( df ∧ iφ ) + } forms a basis of H A . It suffices to show that Π A ( idf ∧ φ ) + = 0 . Suppose this fails to hold. Then (5.6) shows that Π A ( idg ∧ φ ) + = 0 aswell. Let’s write A = d + a with a = q · idg + q · idf .Then for any φ ∈ C = h φ , iφ i , wehave Π A ( a ∧ φ ) + = 0 . Then we get a non-zero complex -form ψ so that d + A ψ = ( a ∧ φ ) + .Equivalently, we have d + A + φ ( a + ψ ) = 0 . Since a + ψ = 0 ∈ H A for all ( a, φ ) , so we concludethat ker d + A + φ ∩ H A = 0 for all φ , which violates the transversality.Now with respect to ordered bases { idg, idf } of i R ⊕ i R ⊂ H A and { Π A ( df ∧ φ ) + , Π A ( df ∧ iφ ) + } of H A , (5.4) and (5.6) tell us the map f considered in the end of subsection 4.2 takesthe form f = (cid:18) ˙ r (0) 0˙ s (0) 1 (cid:19) up to a positive scale. Note that det f = ˙ r (0) . Thus the sign of [ A ] as a bifurcation point isgiven by the sign of ˙ r (0) . This completes the proof. (cid:3) QUIVARIANT TORUS SIGNATURE AND PERIODIC RHO INVARIANT 29
Now we can complete the proof of Theorem 1.1. Since the first homology of V = M ∪ T D × T is generated by [ µ ] and [ γ ] , each flat connection A † t on M extends uniquely to one on V upto U (1) -gauge transformations. We write N = D × T and N o = ( −∞ , × T ∪ D × T .Let’s denoted this new path of connections by ( A † ,Vt ) , and its restriction on D × T by ( A † ,Nt ) .We note that ker Q A † ,Nt ,z ( N o ) = H ( D × T , A † ,Nt,z ) , where A † ,Nt,z is the flat connection on D × T whose holonomies around µ and γ are givenrespectively by e − πitα and z , since under the -surgery gluing map the fundamental groupof D × T is generated by [ µ ] and [ γ ] . For all ( t, z ) ∈ [0 , × S δ , A † ,Nt,z is not the productconnection, thus admits no non-zero twisted harmonic -forms by the Künneth formula. Since ind Q A † ,Nt ,z ( N o ) = 0 , we conclude that Q A † ,Nt ,z ( N o ) : L → L is invertible for all ( t, z ) ∈ [0 , × S δ . Now we can apply the gluing property of the periodic spectral flow Proposition 3.9to conclude that e Sf( Q A † t ( M o , π )) = e Sf( Q A † ,Vt ( V, π )) . Since A † ,V is the product connection and A † ,V corresponds to the representation ϕ α : π ( V ) → U (1) , we can apply Proposition 3.10 to get − ˜ ρ ϕ α ( V, Q ) = e Sf( Q A † ,Vt ( V, π )) , which completes the proof of Theorem 1.1.5.2. Proof of Theorem 1.2.
We prove Theorem 1.2 in this section. Let α ∈ (0 , / be a holonomy parameter whoseassociated representation ϕ α : π ( V ) → U (1) is admissible. To be consistent with the notationsused in subsection 5.1, we let A † α be the flat connection on the trivial line bundle over V whoseholonomy is given by ϕ α . Since ϕ α is admissible, we pick a cross-section Y ⊂ V satisfying(5.7) H ( Y, C ϕ α ) = H ( Y, C ϕ α ) = 0 . In [Rub20], the signature invariant is defined to be σ α ( X, T ) := ρ ϕ α ( Y ) , where ρ ϕ α ( Y ) is the rho invariant of the signature operator L ′ B † α (3.5) which was introducedin [APS75b]. Formally it’s defined to be the difference of eta invariant: ρ ϕ α ( Y ) = η α ( Y ) − η ( Y ) , where η α is the eta invariant associated to L ′ B † α and η ( Y ) is half the eta invariant of thesignature operator. Topologically, if one can find a compact -manifold M ′ with ∂M ′ = Y and ϕ ′ α = ϕ α | π ( Y ) extends to π ( M ′ ) , we have ρ α ( Y ) = σ ( M ′ ) − σ ϕ ′ α ( M ′ ) , where σ means to take the signature of the corresponding intersection form. It is proved in[Rub20] that ρ ϕ α ( Y ) is independent of the choice of the cross-section Y ⊂ V when α is of prime power order, i.e. α = q/p r for p a prime number. Since ρ ϕ α ( Y ) is locally constant, thisinvariant is well-defined for all parameter α avoiding the jumps. Using the Atiyah-Patodi-Singertheory, it’s easy to see that any α satisfying the admissibility condition (5.7) is not a jumpingpoint. Thus we get a well-defined signature invariant σ α ( X, T ) .To compare ˜ ρ ϕ α ( V, Q ) and ρ ϕ α ( Y ) , we consider the manifold Z = ( −∞ , × Y ∪ ˜ V + withone cylindrical end and one periodic end. We extend the function ˜ f : ˜ V + → R to a functionover Z , still denoted by ˜ f , so that ˜ f | { t }× Y = − t for all t ≤ − . Let ˜ ϕ α : π ( Z ) → U (1) bethe unitary representation induced from ϕ α and ˜ A † α the flat connection on Z correspondingto ˜ ϕ α . Then Theorem 2.3 implies that(5.8) ind + Q ( Z ) = Z [ − , × Y a ( Q ) − Z Y ω + Z V df ∧ ω − ρ ( V, Q ) + η ( Y ) and(5.9) ind + Q ˜ A † α ( Z ) = Z [ − , × Y a ( Q ) − Z Y ω + Z V df ∧ ω − ρ ( V, Q ϕ α ) + η α ( Y ) . Taking the difference gives us that(5.10) ˜ ρ ϕ α ( V, Q ) + (cid:16) ind + Q ˜ A † α ( Z ) − ind + Q ( Z ) (cid:17) = ρ ϕ α ( Y ) . It follows from Proposition 2.6 and the admissibility condition (5.7) that ind + Q ˜ A † α ( Z ) = 0 . Soit suffices to prove the following lemma to finish the proof. Lemma 5.4.
Let Z = ( −∞ , × Y ∪ ˜ V + . Then we have ind + Q ( Z ) = 0 . Proof.
Let’s choose δ > sufficiently small. Recall we have the ASD DeRham complex ( E δ ( Z ) )over Z . Then the index is given by ind + Q ( Z ) = dim H ( E δ ( Z )) − dim H ( E δ ( Z )) − dim H ( E δ ( Z )) . Since the only constant function on Z with exponential decay on both ends is the zero function,we have dim H ( E δ ( Z )) = 0 .Now let ω ∈ L ,δ ( T ∗ Z ⊗ C ) be a complex -form satisfying d + ω = 0 . Integration by partsimplies that dω = 0 . [Rub20, Lemma 2.2] tells us that H ∗ ( ˜ V ; Z ) is finitely generated. Let’swrite ˜ V [ m,n ] = W m ∪ ... ∪ W n for the interval of periods from m to n with integers m < n . Thenone can find an integer N > such that each class in H ( ˜ V ; Z ) can be represented by cyclesin ˜ V [ i,i + N ] for all i ∈ Z . Let [ c ] ∈ H ( Z ; Z ) . We let c i ⊂ ˜ V [ i,i + N ] be the representative of [ c ] for i ≥ . Then ω · [ c ] = Z c i ω = lim i →∞ Z c i ω = 0 , which shows that ω is compactly supported and exact. Thus dim H ( E δ ( Z )) = 0 .Let ζ ∈ L δ (Λ + T ∗ Z ⊗ C ) be a self-dual -form satisfying e − δ ˜ f d ∗ e δ ˜ f ζ = 0 . Let ζ δ = e δ ˜ f ζ . Since ζ is self-dual, we conclude that dζ δ = 0 and d ∗ ζ δ = 0 . Thus ζ δ is an L -harmonic self-dual -formon Z . Using the same argument as above, we conclude that ζ δ = 0 . Thus dim H ( E δ ( Z )) = 0 .This completes the proof. (cid:3) QUIVARIANT TORUS SIGNATURE AND PERIODIC RHO INVARIANT 31
Computation on Mapping Tori.
In this subsection, we supply an argument for (1.7). For ease of notation, we write Σ for the n -fold branched cover Σ n ( Y, K ) , τ for the covering transformation τ n , and X for the mappingtorus X n ( Y, K ) . We write J ⊂ Σ for the preimage of K , which we used to write K n in theintroduction, and T for the mapping torus of J in X . Let’s write i : Σ → X for the inclusionmap on a slice of the mapping torus.The argument of [RS04, Proposition 3.1] tells us that the irreducible α -representations of ( X, T ) is in two-to-one correspondence with the τ -invariant irreducible α -representations of (Σ , J ) , namely i ∗ : χ α, ∗ ( X, T ) → χ α, ∗ τ (Σ , J ) is a double cover. As pointed out in [Ech19,Section 8], the Zariski tangent spaces and orientations of M α, ∗ σ ( X, T ) can be identified withthose of M α, ∗ σ,τ (Σ , J ) with respect to generic perturbations, where M α, ∗ σ,τ (Σ , J ) means the spaceof irreducible perturbed τ -invariant flat α -connections on (Σ , J ) . Thus it suffices to count M τ, ∗ σ (Σ , J ) . Since the holonomy perturbations preserve the τ -invariance, and away from thesupport of a perturbation each connection in M τ, ∗ σ (Σ , J ) is flat, each connection in M α, ∗ σ,τ (Σ , J ) comes from a perturbed flat connection in ( Y, K ) with respect to certain holonomy parameter α ′ .In [Ech19, Lemma 48], Echeverria proved that the pull-back map p ∗ : M α ′ , ∗ σ ( Y, K ) →M α, ∗ σ,τ (Σ , J ) is either injective or empty given a holonomy parameter α ′ ∈ (0 , / . So thekey point is to determine the holonomy parameters α ′ with respect to which the pull-back mapis non-empty. Let [ A ] = p ∗ [ A ′ ] ∈ M α, ∗ σ,τ (Σ , J ) . The the holonomy of A around the meridian µ J is e − πiα . Thus one can choose representative A ′ of [ A ′ ] such that the holonomy of A ′ aroundthe meridian µ K satisfies(5.11) (cid:0) Hol µ K A ′ (cid:1) n = Hol µ J A = e − πiα = ⇒ Hol µ K A ′ = e − πi ( α + j ) /n , for some j = 0 , ..., n − . Conversely any perturbed flat connection A ′ on Y \ K satisfies (5.11)pulls back to one on Σ \J . Due to the normalization that the holonomy parameter lies in (0 , / , we conclude that there are n such choices parametrized by j = 0 , ..., n − :(5.12) α ′ j = ( ( α + j ) /n if ( α + j ) /n ∈ (0 , / − ( α + j ) /n if ( α + j ) /n ∈ (1 / , Since α ∈ (0 , / , it’s direct to check that all values of α ′ j in (5.12) are distinct. Invoking theresult of Herald [Her97, Theorem 0.1], when e − πiα ′ j is not a root of the Alexander polynomial ∆ ( Y, K ) for each j = 0 , ..., n − , we get M α ′ j , ∗ σ ( Y, K ) = 4 λ ( Y ) + 12 σ α ′ j ( Y, K ) = 4 λ ( Y ) + 12 σ α + j ) /n ( Y, K ) , where the second equality has used the symmetry of the Levine-Tristram invariant [Lin92,Proposition 2.3] in the case when ( α + j ) /n ∈ (1 / , . Thus we arrive at the claimed compu-tation(5.13) λ F O ( X, T , α ) = 2 n − X j =0 M α ′ j , ∗ σ ( Y, K ) = 8 nλ ( Y ) + n − X j =0 σ α + j ) /n ( Y, K ) . References [APS75a] M. F. Atiyah, V. K. Patodi, and I. M. Singer. Spectral asymmetry and Riemannian geometry. I.
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